ROUND ROCK, Texas — Michael Dell believes he can revive the company bearing his name if his group of investors can buy it for $24.4 billion. But that deal is in danger of falling apart, increasing the chances that the personal computer giant’s founder might not be CEO much longer.

The formidable challenges already facing Dell and its CEO got more daunting Thursday with the slumping company’s decision to delay a vote on Michael Dell’s proposed buyout.

The postponement until next Wednesday signals that a five-month campaign by Michael Dell and the company’s board hasn’t overcome the staunch resistance of billionaire Carl Icahn and other shareholders, who argue the buyout price is too low and discounts the company’s long-term prospects.

Michael Dell’s offer has the support of three shareholder-advisory firms and is backed by the financial clout of buyout specialist Silver Lake Partners and a group of lenders. It would take the company private so that Dell can try to engineer a long-term recovery without the glare of Wall Street and its fixation with quarter-to-quarter expectations.

The offer works out to $13.65 per share, or more than 40 percent below where the stock stood in early 2007. That was when Michael Dell returned for a second stint as the company’s CEO — just a few months before Apple started selling the iPhone, which triggered a mobile-computing revolution that left Dell on shaky ground.

Rather than relying on the types of desktops and laptops made by Dell, more people are embracing convenient and powerful smartphones and tablet computers to connect to the Internet and handle other common computing tasks. In a telltale sign, tablets are expected to outsell laptops for the first time this year.